Answer:
To find the curvature of the curve defined by the vector-valued function r(t) = 3ti + etj + e^(-tk) at the point P(0, 1, 1), we'll need to compute several derivatives and then use the curvature formula:
κ = |r'(t) × r''(t)| / |r'(t)|^3
Let's start by finding the derivatives:
1. r(t) = 3ti + etj + e^(-tk)
2. r'(t) = (3i + ej - e^(-tk)k)
3. r''(t) = (0i + 0j + e^(-tk)k^2)
Now, plug these derivatives into the curvature formula:
r'(0) = 3i + ej - ek
r''(0) = 0i + 0j - 0k
| r'(0) × r''(0) | = |(3i + ej - ek) × (0i + 0j - 0k)| = |(3i + ej - ek) × 0k| = 0
|r'(0)|^3 = |3i + ej - ek|^3 = √(3^2 + 1^2 + 1^2)^3 = √11^3 = 11^(3/2)
Now, plug these values into the curvature formula:
κ = | r'(0) × r''(0) | / | r'(0) |^3
κ = 0 / (11^(3/2))
κ = 0
So, the curvature of the curve at the point P(0, 1, 1) is κ = 0.
Explanation: